Factor the following expression: $-3$ $x^2$ $-5$ $x+$ $12$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-3)}{(12)} &=& -36 \\ {a} + {b} &=& & & {-5} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-36$ and add them together. Remember, since $-36$ is negative, one of the factors must be negative. The factors that add up to ${-5}$ will be your ${a}$ and ${b}$ When ${a}$ is ${4}$ and ${b}$ is ${-9}$ $ \begin{eqnarray} {ab} &=& ({4})({-9}) &=& -36 \\ {a} + {b} &=& {4} + {-9} &=& -5 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-3}x^2 +{4}x {-9}x +{12} $ Group the terms so that there is a common factor in each group: $ ({-3}x^2 +{4}x) + ({-9}x +{12}) $ Factor out the common factors: $ x(-3x + 4) + 3(-3x + 4) $ Notice how $(-3x + 4)$ has become a common factor. Factor this out to find the answer. $(-3x + 4)(x + 3)$